# Solve Poisson Equation on ring-shaped domain

Being my first question in Math StackExchange, a difficulty arises when I attempted to solve a poisson equation on a ring-shaped domain

$$\begin{cases} \triangle u = 12(x^2 - y^2),\quad u \in \Omega\\ u(x,y) = 1, \quad x^2 + y^2 = a^2,\\ \dfrac{\partial u}{\partial n} =0, \quad x^2 + y^2 = b^2\end{cases}$$ in which $$\Omega := \{(x,y)| a^2 \leq x^2 + y^2 \leq b^2\}$$,with real number $$0 < a < b$$.

Naturally I tried to use Separation of Variables, which indicates the ansatz $$u(x,y) = R(r)\Theta(\theta)$$ and in polar coordinates we have $$\triangle u(r,\theta) = \dfrac{\partial^2 u}{\partial r^2} + \dfrac{1}{r} \dfrac{\partial u}{\partial r} + \dfrac{1}{r^2} \dfrac{\partial^2 u}{\partial \theta^2}$$ Thus we get the equation $$R''(r)\Theta(\theta) + \dfrac{1}{r}R'(r)\Theta(\theta) + \dfrac{1} {r^2}R(r)\Theta''(\theta) = 12 r^2 \cos 2\theta$$ by taking $$R(r) = Ar^4$$ we have $$A[\Theta''(\theta) + 16\Theta(\theta)] = 12 \cos 2\theta$$ and I find $$\Theta(\theta) = 1/A \cos 2\theta$$ a natural soluion.

However, by giving the solution $$u(r,\theta) = r^4\cos 2\theta$$ , I find it hard to imagine a satisfaction of boundary condition $$u\bigg|_{x^2 + y^2 = a^2} =1$$,which is irrelevant to angle function $$\Theta(\theta)$$ and on which symmetry holds with $$\theta$$.

It is possible that the ansatz should be improved, but I have no idea about that. Any comments or suggestions will be greatly appreciated.

Update:

I think I've found the particular solution $$u_p = r^4\cos 2\theta$$, and a solution for homogeneous equation should be added. let $$u = u_p + \sum_{k=1}^{+\infty} R_k(r)\Theta_k(\theta)$$ ,where the series expansion solves the Laplace equation.

By separation of Variable I get $$r^2 R_k'' + rR_k' - R_k\lambda_k^2 = 0,\quad \Theta_k'' + \lambda_k^2\Theta_k = 0$$ By solving the Euler-type equation and the harmonic oscillation equation I get $$R_k = A_k^1 r^{\lambda_k} + A_k^2 r^{-\lambda_k},\quad \Theta_k = C_k^1 \cos \lambda_k \theta + C_k^2 \sin \lambda_k \theta$$ Can this series expansion satisfy the two boundary conditions?

• I think assuming a separated solution is already going too far. The inhomogeneity of the equation as well as the weird geometry and BCs make me think that this equation might not even have a closed form. Commented Dec 26, 2021 at 18:57
• As @Chee Han points out, solutions to the homogeneous equation $\frac{\partial^{2}u}{\partial r^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}u}{\partial \theta^{2}}=0$ can be added on to your particular solution. For example $u(r,\theta)=1$ or $u(r,\theta)=-a^{2}r^{2}\cos(2\theta)$. In fact with $u(r,\theta)=1+\left(r^{4}-a^{2}r^{2}\right)\cos(2\theta)$ you have a solution that satisfies your first boundary condition though not, I think, your second.
– Ali
Commented Dec 27, 2021 at 0:08
• yeah@Ali ,this weied boundary condition really make me confused. Commented Dec 27, 2021 at 0:08
• As you may have worked out by now, the answer is $u(r,\theta)=1 +\left(r^{4}+s r^{2} + t r^{-2}\right)\cos(2\theta)$ where $\begin{pmatrix}a^{2}&a^{-2}\\b&-b^{-3}\end{pmatrix} \begin{pmatrix}s\\t\end{pmatrix}= \begin{pmatrix}-a^{4}\\-2b^{3}\end{pmatrix}$. @K.defaoite was too pessimistic! Of course you'd never meet such a problem in the wild.
– Ali
Commented Dec 27, 2021 at 9:39

I myself make this mistake countless of time! The general solution to an inhomogeneous linear PDE (or ODE) has two parts: the homogeneous solution $$u_h$$ and the particular solution $$u_p$$. What you found is actually "the particular solution", you still need to find the constant $$A$$ which can be found but choosing $$\Theta = \cos(2\theta)$$ in your equation above.
• Can you be more explicit please? I don't understand the role you see for $A$. It seems to be eliminated almost as soon as it is introduced.
• @Amelius258 I didn't actually try and solve it with the boundary condition, did it really not work? By the way, I think I got $A = 16/21$. Commented Dec 27, 2021 at 0:13